Kindly solve appropriately

Five years ago, an insurance company began to issue insurance policies covering medical expenses for dogs. The insurance company classifies dogs into three risk categories: large pedigree (category 1), small pedigree (category 2) and non-pedigree (category 3). The number of claims n,, in the ith category in the jth year is assumed to have a Poisson distribution with unknown parameter 9, Data on the number of claims in each category over the last 5 years is set out as follows: Year 1 2 5 2 Category 1 30 43 49 58 GO 240 12,144 Category 2 37 49 58 52 64 260 13,934 Category 3 26 31 18 37 32 144 4.354 Prior beliefs about 0, are given by a gamma distribution with mean 50 and variance 25. (i) Find the Bayes estimate of 0, under quadratic loss. 15] (ii) Calculate the expected claims for year 6 of each category under the assumptions of Empirical Bayes Credibility Theory Model 1 [6] (i1) Explain the main differences between the approach in (1) and that in (i1). [3] (iv) Explain why the assumption of a Poisson distribution with a constant parameter may not be appropriate and describe how each approach might be generalised. 131 [Total 17]